Nonlinear functional analysis

Nonlinear Functional Analysis Gerald Teschl Gerald Teschl Fakultăat fă ur Mathematik Nordbergstraòe 15 Universităat Wien 1090 Wien, Austria E-mail address: Gerald.Teschl@univie.ac.at URL: http://www.mat.univie.ac.at/~gerald/ 1991 Mathematics subject classification 46-01, 47H10, 47H11, 58Fxx, 76D05 Abstract This manuscript provides a brief introduction to nonlinear functional analysis We start out with calculus in Banach spaces, review differentiation and integration, derive the implicit function theorem (using the uniform contraction principle) and apply the result to prove existence and uniqueness of solutions for ordinary differential equations in Banach spaces Next we introduce the mapping degree in both finite (Brouwer degree) and infinite dimensional (Leray-Schauder degree) Banach spaces Several applications to game theory, integral equations, and ordinary differential equations are discussed As an application we consider partial differential equations and prove existence and uniqueness for solutions of the stationary Navier-Stokes equation Finally, we give a brief discussion of monotone operators Keywords and phrases Mapping degree, fixed-point theorems, differential equations, Navier–Stokes equation Typeset by LATEX and Makeindex Version: October 13, 2005 Copyright c 1998-2004 by Gerald Teschl ii Preface The present manuscript was written for my course Nonlinear Functional Analysis held at the University of Vienna in Summer 1998 and 2001 It is supposed to give a brief introduction to the field of Nonlinear Functional Analysis with emphasis on applications and examples The material covered is highly selective and many important and interesting topics are not covered It is available from http://www.mat.univie.ac.at/~gerald/ftp/book-nlfa/ Acknowledgments I’d like to thank Volker Enß for making his lecture notes available to me and Matthias Hammerl for pointing out errors in previous versions Gerald Teschl Vienna, Austria February 2001 iii iv Preface Contents Preface iii Analysis in Banach spaces 1.1 Differentiation and integration in Banach spaces 1.2 Contraction principles 1.3 Ordinary differential equations 1 The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Brouwer mapping degree Introduction Definition of the mapping degree and the determinant formula Extension of the determinant formula The Brouwer fixed-point theorem Kakutani’s fixed-point theorem and applications to game theory Further properties of the degree The Jordan curve theorem The 3.1 3.2 3.3 3.4 3.5 Leray–Schauder mapping degree The mapping degree on finite dimensional Banach spaces Compact operators The Leray–Schauder mapping degree The Leray–Schauder principle and the Schauder fixed-point Applications to integral and differential equations The 4.1 4.2 4.3 stationary Navier–Stokes equation 43 Introduction and motivation 43 An insert on Sobolev spaces 44 Existence and uniqueness of solutions 50 v 11 11 13 17 24 25 29 31 theorem 33 33 34 35 37 39 vi Contents Monotone operators 53 5.1 Monotone operators 53 5.2 The nonlinear Lax–Milgram theorem 55 5.3 The main theorem of monotone operators 57 Bibliography 61 Glossary of notations 63 Index 65 Chapter Analysis in Banach spaces 1.1 Differentiation and integration in Banach spaces We first review some basic facts from calculus in Banach spaces Let X and Y be two Banach spaces and denote by C(X, Y ) the set of continuous functions from X to Y and by L(X, Y ) ⊂ C(X, Y ) the set of (bounded) linear functions Let U be an open subset of X Then a function F : U → Y is called differentiable at x ∈ U if there exists a linear function dF (x) ∈ L(X, Y ) such that F (x + u) = F (x) + dF (x) u + o(u), (1.1) where o, O are the Landau symbols The linear map dF (x) is called derivative of F at x If F is differentiable for all x ∈ U we call F differentiable In this case we get a map dF : U → L(X, Y ) (1.2) x → dF (x) If dF is continuous, we call F continuously differentiable and write F ∈ C (U, Y ) Let Y = m j=1 Yj and let F : X → Y be given by F = (F1 , , Fm ) with Fj : X → Yi Then F ∈ C (X, Y ) if and only if Fj ∈ C (X, Yj ), ≤ j ≤ m, and in this case dF = (dF1 , , dFm ) Similarly, if X = m i=1 Xi , then one can define the partial derivative ∂i F ∈ L(Xi , Y ), which is the derivative of F considered as a function of the i-th variable alone (the other variables being fixed) We have dF v = ni=1 ∂i F vi , v = (v1 , , ) ∈ X, and F ∈ C (X, Y ) if and only if all partial derivatives exist and are continuous Chapter Analysis in Banach spaces In the case of X = Rm and Y = Rn ,the matrix representation of dF with respect to the canonical basis in Rm and Rn is given by the partial derivatives ∂i Fj (x) and is called Jacobi matrix of F at x We can iterate the procedure of differentiation and write F ∈ C r (U, Y ), r ≥ 1, if the r-th derivative of F , dr F (i.e., the derivative of the (r − 1)-th derivative of F ), exists and is continuous Finally, we set C ∞ (U, Y ) = r∈N C r (U, Y ) and, for notational convenience, C (U, Y ) = C(U, Y ) and d0 F = F It is often necessary to equip C r (U, Y ) with a norm A suitable choice is |F | = max sup |dj F (x)| (1.3) 0≤j≤r x∈U The set of all r times continuously differentiable functions for which this norm is finite forms a Banach space which is denoted by Cbr (U, Y ) If F is bijective and F , F −1 are both of class C r , r ≥ 1, then F is called a diffeomorphism of class C r Note that if F ∈ L(X, Y ), then dF (x) = F (independent of x) and dr F (x) = 0, r > For the composition of mappings we note the following result (which is easy to prove) Lemma 1.1 (Chain rule) Let F ∈ C r (X, Y ) and G ∈ C r (Y, Z), r ≥ Then G ◦ F ∈ C r (X, Z) and d(G ◦ F )(x) = dG(F (x)) ◦ dF (x), x ∈ X (1.4) In particular, if λ ∈ Y ∗ is a linear functional, then d(λ ◦ F ) = dλ ◦ dF = λ ◦ dF In addition, we have the following mean value theorem Theorem 1.2 (Mean value) Suppose U ⊆ X and F ∈ C (U, Y ) If U is convex, then |F (x) − F (y)| ≤ M |x − y|, M = max |dF ((1 − t)x + ty)| (1.5) 0≤t≤1 Conversely, (for any open U ) if |F (x) − F (y)| ≤ M |x − y|, x, y ∈ U, (1.6) then sup |dF (x)| ≤ M x∈U (1.7) 1.1 Differentiation and integration in Banach spaces Proof Abbreviate f (t) = F ((1 − t)x + ty), ≤ t ≤ 1, and hence df (t) = ˜ = M |x − y| For the first part it dF ((1 − t)x + ty)(y − x) implying |df (t)| ≤ M suffices to show ˜ + δ)t ≤ φ(t) = |f (t) − f (0)| − (M (1.8) for any δ > Let t0 = max{t ∈ [0, 1]|φ(t) ≤ 0} If t0 < then φ(t0 + ε) = ≤ ≤ ≤ ˜ + δ)(t0 + ε) |f (t0 + ε) − f (t0 ) + f (t0 ) − f (0)| − (M ˜ + δ)ε + φ(t0 ) |f (t0 + ε) − f (t0 )| − (M ˜ + δ)ε |df (t0 )ε + o(ε)| − (M ˜ + o(1) − M ˜ − δ)ε = (−δ + o(1))ε ≤ 0, (M (1.9) for ε ≥ 0, small enough Thus t0 = To prove the second claim suppose there is an x0 ∈ U such that |dF (x0 )| = M + δ, δ > Then we can find an e ∈ X, |e| = such that |dF (x0 )e| = M + δ and hence M ε ≥ |F (x0 + εe) − F (x0 )| = |dF (x0 )(εe) + o(ε)| ≥ (M + δ)ε − |o(ε)| > M ε since we can assume |o(ε)| < εδ for ε > small enough, a contradiction (1.10) ✷ As an immediate consequence we obtain Corollary 1.3 Suppose U is a connected subset of a Banach space X A mapping F ∈ C (U, Y ) is constant if and only if dF = In addition, if F1,2 ∈ C (U, Y ) and dF1 = dF2 , then F1 and F2 differ only by a constant Next we want to look at higher derivatives more closely Let X = m i=1 Xi , then F : X → Y is called multilinear if it is linear with respect to each argument It is not hard to see that F is continuous if and only if |F | = sup x: Qm i=1 |F (x1 , , xm )| < ∞ (1.11) |xi |=1 If we take n copies of the same space, the set of multilinear functions F : X n → Y will be denoted by Ln (X, Y ) A multilinear function is called symmetric provided its value remains unchanged if any two arguments are switched With the norm from above it is a Banach space and in fact there is a canonical isometric isomorphism between Ln (X, Y ) and L(X, Ln−1 (X, Y )) given by F : (x1 , , xn ) → Chapter Monotone operators 5.1 Monotone operators The Leray–Schauder theory can only be applied to compact perturbations of the identity If F is not compact, we need different tools In this section we briefly present another class of operators, namely monotone ones, which allow some progress If F : R → R is continuous and we want F (x) = y to have a unique solution for every y ∈ R, then f should clearly be strictly monotone increasing (or decreasing) and satisfy limx→±∞ F (x) = ±∞ Rewriting these conditions slightly such that they make sense for vector valued functions the analogous result holds Lemma 5.1 Suppose F : Rn → Rn is continuous and satisfies F (x)x = ∞ |x|→∞ |x| (5.1) F (x) = y (5.2) lim Then the equation has a solution for every y ∈ Rn If F is strictly monotone (F (x) − F (y))(x − y) > 0, x = y, (5.3) then this solution is unique Proof Our first assumption implies that G(x) = F (x) − y satisfies G(x)x = F (x)x − yx > for |x| sufficiently large Hence the first claim follows from Theorem 2.13 The second claim is trivial ✷ 53 54 Chapter Monotone operators Now we want to generalize this result to infinite dimensional spaces Throughout this chapter, X will be a Hilbert space with scalar product , An operator F : X → X is called monotone if F (x) − F (y), x − y ≥ 0, x, y ∈ X, (5.4) x = y ∈ X, (5.5) strictly monotone if F (x) − F (y), x − y > 0, and finally strongly monotone if there is a constant C > such that F (x) − F (y), x − y ≥ C|x − y|2 , x, y ∈ X (5.6) Note that the same definitions can be made if X is a Banach space and F : X → X ∗ Observe that if F is strongly monotone, then it automatically satisfies lim |x|→∞ F (x), x = ∞ |x| (5.7) (Just take y = in the definition of strong monotonicity.) Hence the following result is not surprising Theorem 5.2 (Zarantonello) Suppose F ∈ C(X, X) is (globally) Lipschitz continuous and strongly monotone Then, for each y ∈ X the equation (5.8) F (x) = y has a unique solution x ∈ X Proof Set G(x) = x − t(F (x) − y), t > 0, (5.9) then F (x) = y is equivalent to the fixed point equation G(x) = x (5.10) It remains to show that G is a contraction We compute |G(x) − G(˜ x)|2 = |x − x˜|2 − 2t F (x) − F (˜ x), x − x˜ + t2 |F (x) − F (˜ x)|2 C (5.11) ≤ (1 − (Lt) + (Lt)2 )|x − x˜|2 , L 5.2 The nonlinear Lax–Milgram theorem 55 where L is a Lipschitz constant for F (i.e., |F (x) − F (˜ x)| ≤ L|x − x˜|) Thus, if t ∈ 2C (0, L ), G is a contraction and the rest follows from the contraction principle ✷ Again observe that our proof is constructive In fact, the best choice for t is clearly t = CL such that the contraction constant θ = − ( CL )2 is minimal Then the sequence C xn+1 = xn − (1 − ( )2 )(F (xn ) − y), x0 = x, (5.12) L converges to the solution 5.2 The nonlinear Lax–Milgram theorem As a consequence of the last theorem we obtain a nonlinear version of the Lax– Milgram theorem We want to investigate the following problem: a(x, y) = b(y), for all y ∈ X, (5.13) where a : X → R and b : X → R For this equation the following result holds Theorem 5.3 (Nonlinear Lax–Milgram theorem) Suppose b ∈ L(X, R) and a(x, ) ∈ L(X, R), x ∈ X, are linear functionals such that there are positive constants L and C such that for all x, y, z ∈ X we have a(x, x − y) − a(y, x − y) ≥ C|x − y|2 (5.14) |a(x, z) − a(y, z)| ≤ L|z||x − y| (5.15) and Then there is a unique x ∈ X such that (5.13) holds Proof By the Riez theorem there are elements F (x) ∈ X and z ∈ X such that a(x, y) = b(y) is equivalent to F (x) − z, y = 0, y ∈ X, and hence to F (x) = z (5.16) By (5.14) the operator F is strongly monotone Moreover, by (5.15) we infer |F (x) − F (y)| = sup | F (x) − F (y), x˜ | ≤ L|x − y| (5.17) x ˜∈X,|˜ x|=1 that F is Lipschitz continuous Now apply Theorem 5.2 ✷ 56 Chapter Monotone operators The special case where a ∈ L2 (X, R) is a bounded bilinear form which is strongly continuous, that is, a(x, x) ≥ C|x|2 , x ∈ X, (5.18) is usually known as (linear) Lax–Milgram theorem The typical application of this theorem is the existence of a unique weak solution of the Dirichlet problem for elliptic equations x ∈ U, x ∈ ∂U, ∂i Aij (x)∂j u(x) + bj (x)∂j u(x) + c(x)u(x) = f (x), u(x) = 0, (5.19) where U is a bounded open subset of Rn By elliptic we mean that all coefficients A, b, c plus the right hand side f are bounded and a0 > 0, where a0 = inf e∈S n ,x∈U ei Aij (x)ej , b0 = − inf b(x), x∈U c0 = inf c(x) x∈U (5.20) As in Section 4.3 we pick H01 (U, R) with scalar product u, v = (5.21) (∂j u)(∂j v)dx U as underlying Hilbert space Next we multiply (5.19) by v ∈ H01 and integrate over U f (x)v(x) dx (5.22) ∂i Aij (x)∂j u(x) + bj (x)∂j u(x) + c(x)u(x) v(x) dx = U U After a partial integration we can write this equation as a(v, u) = f (v), v ∈ H01 , (5.23) where a(v, u) = ∂i v(x)Aij (x)∂j u(x) + bj (x)v(x)∂j u(x) + c(x)v(x)u(x) dx U f (v) = f (x)v(x) dx, (5.24) U We call a solution of (5.23) a weak solution of the elliptic Dirichlet problem (5.19) 5.3 The main theorem of monotone operators 57 By a simple use of the Cauchy-Schwarz and Poincar´e-Friedrichs inequalities we see that the bilinear form a(u, v) is bounded To be able to apply the (linear) Lax–Milgram theorem we need to show that it satisfies a(u, u) ≥ |∂j u|2 dx Using (5.20) we have a0 |∂j u|2 − b0 |u||∂j u| + c0 |u|2 , a(u, u) ≥ (5.25) U where −b0 = inf b(x), c0 = inf c(x) and we need to control the middle term If b0 ≤ there is nothing to and it suffices to require c0 ≥ If b0 > we distribute the middle term by means of the elementary inequality ε |u||∂j u| ≤ |u|2 + |∂j u|2 2ε (5.26) which gives a(u, u) ≥ (a0 − U b0 εb0 )|∂j u|2 + (c0 − )|u|2 2ε (5.27) b0 b0 Since we need a0 − 2ε > and c0 − εb20 ≥ 0, or equivalently 2cb00 ≥ ε > 2a , we see that we can apply the Lax–Milgram theorem if 4a0 c0 > b0 In summary, we have proven Theorem 5.4 The elliptic Dirichlet problem (5.19) has a unique weak solution u ∈ H01 (U, R) if a0 > 0, b0 ≤ 0, c0 ≥ or 4a0 c0 > b20 5.3 The main theorem of monotone operators Now we return to the investigation of F (x) = y and weaken the conditions of Theorem 5.2 We will assume that X is a separable Hilbert space and that F : X → X is a continuous monotone operator satisfying lim |x|→∞ F (x), x = ∞ |x| (5.28) In fact, if suffices to assume that F is weakly continuous lim F (xn ), y = F (x), y , n→∞ whenever xn → x for all y ∈ X (5.29) 58 Chapter Monotone operators The idea is as follows: Start with a finite dimensional subspace Xn ⊂ X and project the equation F (x) = y to Xn resulting in an equation Fn (xn ) = yn , xn , yn ∈ Xn (5.30) More precisely, let Pn be the (linear) projection onto Xn and set Fn (xn ) = Pn F (xn ), yn = Pn y (verify that Fn is continuous and monotone!) Now Lemma 5.1 ensures that there exists a solution un Now chose the subspaces Xn such that Xn → X (i.e., Xn ⊂ Xn+1 and ∞ n=1 Xn is dense) Then our hope is that un converges to a solution u This approach is quite common when solving equations in infinite dimensional spaces and is known as Galerkin approximation It can often be used for numerical computations and the right choice of the spaces Xn will have a significant impact on the quality of the approximation So how should we show that xn converges? First of all observe that our construction of xn shows that xn lies in some ball with radius Rn , which is chosen such that Fn (x), x > |yn ||x|, |x| ≥ Rn , x ∈ Xn (5.31) Since Fn (x), x = Pn F (x), x = F (x), Pn x = F (x), x for x ∈ Xn we can drop all n’s to obtain a constant R which works for all n So the sequence xn is uniformly bounded (5.32) |xn | ≤ R Now by a well-known result there exists a weakly convergent subsequence That is, after dropping some terms, we can assume that there is some x such that xn x, that is, (5.33) xn , z → x, z , for every z ∈ X And it remains to show that x is indeed a solution This follows from Lemma 5.5 Suppose F : X → X is weakly continuous and monotone, then y − F (z), x − z ≥ for every z ∈ X (5.34) implies F (x) = y Proof Choose z = x ± tw, then ∓ y − F (x ± tw), w ≥ and by continuity ∓ y − F (x), w ≥ Thus y − F (x), w = for every w implying y − F (x) = ✷ Now we can show 5.3 The main theorem of monotone operators 59 Theorem 5.6 (Browder, Minty) Suppose F : X → X is weakly continuous, monotone, and satisfies F (x), x lim = ∞ (5.35) |x|→∞ |x| Then the equation F (x) = y (5.36) has a solution for every y ∈ X If F is strictly monotone then this solution is unique Proof Abbreviate yn = F (xn ), then we have y − F (z), xn − z = yn − Fn (z), xn − z ≥ for z ∈ Xn Taking the limit implies y − F (z), x − z ≥ for every z ∈ X∞ = ∞ n=1 Xn Since X∞ is dense, y − F (z), x − z ≥ for every z ∈ X by continuity and hence F (x) = y by our lemma ✷ Note that in the infinite dimensional case we need monotonicity even to show existence Moreover, this result can be further generalized in two more ways First of all, the Hilbert space X can be replaced by a reflexive Banach space if F : X → X ∗ The proof is almost identical Secondly, it suffices if t → F (x + ty), z (5.37) is continuous for t ∈ [0, 1] and all x, y, z ∈ X, since this condition together with monotonicity can be shown to imply weak continuity 60 Chapter Monotone operators Bibliography [1] M Berger and M Berger, Perspectives in Nonlinearity, Benjamin, New York, 1968 [2] L C Evans, Weak Convergence Methods for nonlinear Partial Differential Equations, CBMS 74, American Mathematical Society, Providence, 1990 [3] S.-N Chow and J K Hale, Methods of Bifurcation Theory, Springer, New York, 1982 [4] K Deimling, Nichtlineare Gleichungen und Abbildungsgrade, Springer, Berlin, 1974 [5] K Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985 [6] J Franklin, Methods of Mathematical Economics, Springer, New York 1980 [7] O.A Ladyzhenskaya, The Boundary Values Problems of Mathematical Physics, Springer, New York, 1985 [8] N Lloyd, Degree Theory, Cambridge University Press, London, 1978 [9] J.J Rotman, Introduction to Algebraic Topology, Springer, New York, 1988 [10] M Ruˇ ˙ ziˇcka, Nichtlineare Funktionalanalysis, Springer, Berlin, 2004 [11] E Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Springer, New York 1995 [12] E Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Springer, New York 1995 61 62 Bibliography 63 64 Glossary of notations Glossary of notations Bρ (x) conv(.) C(U, Y ) C r (U, Y ) C0r (U, Y ) C(U, Y ) CP(f ) CS(K) CV(f ) deg(D, f, y) det dim div dist(U, V ) Dyr (U, Y ) dF F(X, Y ) GL(n) H(C) H (U, Rn ) H01 (U, Rn ) inf Jf (x) L(X, Y ) Lp (U, Rn ) max n(γ, z0 ) O(.) o(.) ball of radius ρ around x convex hull set of continuous functions from U to Y , set of r times continuously differentiable functions, functions in C r with compact support, 45 set of compact functions from U to Y , 34 critical points of f , 13 nonempty convex subsets of K, 26 critical values of f , 13 mapping degree, 13, 22 determinant dimension of a linear space divergence = inf (x,y)∈U ×V |x − y| distance of two sets functions in C r (U , Y ) which not attain y on the boundary derivative of F , set of compact finite dimensional functions, 34 general linear group in n dimensions set of holomorphic functions, 11 Sobolev space, 45 Sobolev space, 45 infimum = det f (x) Jacobi determinant of f at x, 13 set of bounded linear functions, Lebesgue space of p integrable functions, 44 maximum winding number Landau symbol, f = O(g) iff lim supx→x0 |f (x)/g(x)| < ∞ Landau symbol, f = o(g) iff limx→x0 |f (x)/g(x)| = Glossary of notations ∂U ∂x F (x, y) RV(f ) R(I, X) S(I, X) sgn sup supp boundary of the set U partial derivative with respect to x, regular values of f , 13 set of regulated functions, set of simple functions, sign of a number supremum support of a functions 65 Index Arzel`a-Ascoli theorem, 40 Functional, linear, Best reply, 27 Brouwer fixed-point theorem, 24 Galerkin approximation, 58 Gronwall’s inequality, 41 Chain rule, Characteristic function, Compact operator, 34 Contraction principle, Critical values, 13 Holomorphic function, 11 Homotopy, 12 Homotopy invariance, 13 Implicit function theorem, Integral, Integration by parts, 45 Inverse function theorem, Derivative, partial, Diffeomorphism, Differentiable, Differential equations, Distribution, 46 Jordan curve theorem, 31 Kakutani’s fixed-point theorem, 26 Ladyzhenskaya inequality, 48 Landau symbols, Lax–Milgram theorem, 55 Leray–Schauder principle, 37 Elliptic equation, 56 Embedding, 48 Equilibrium Nash, 28 Mean value theorem, monotone, 54 operator, 53 strictly, 54 strongly, 54 Multilinear function, Finite dimensional operator, 34 Fixed-point theorem Altman, 38 Brouwer, 24 contraction principle, Kakutani, 26 Krasnosel’skii, 38 Rothe, 38 Schauder, 37 Nash equilibrium, 28 Nash theorem, 28 Navier–Stokes equation, 44 66 Index stationary, 44 n-person game, 27 Payoff, 27 Peano theorem, 40 Poincar´e inequality, 47 Poincar´e-Friedrichs inequality, 46 Prisoners dilemma, 28 Proper, 35 Reduction property, 29 Regular values, 13 Regulated function, Rellich’s compactness theorem, 48 Rouch´es theorem, 12 Sard’s theorem, 17 Simple function, Stokes theorem, 19 Strategy, 27 Symmetric multilinear function, Uniform contraction principle, Weak solution, 50, 56 Winding number, 11 67 ... for my course Nonlinear Functional Analysis held at the University of Vienna in Summer 1998 and 2001 It is supposed to give a brief introduction to the field of Nonlinear Functional Analysis with... 47H10, 47H11, 58Fxx, 76D05 Abstract This manuscript provides a brief introduction to nonlinear functional analysis We start out with calculus in Banach spaces, review differentiation and integration,... 53 5.2 The nonlinear Lax–Milgram theorem 55 5.3 The main theorem of monotone operators 57 Bibliography 61 Glossary of notations 63 Index 65 Chapter Analysis in